On partition functions of refined Chern-Simons theories on $S^3$

TL;DR

This paper introduces a new explicit expression for the partition function of refined Chern-Simons theory on $S^3$, generalizing known representations to all gauge algebras and facilitating duality with refined topological strings.

Contribution

It provides a novel explicit form of the partition function for refined Chern-Simons theory applicable to all gauge groups, extending previous results and enabling new duality analyses.

Findings

Partition function equals 1 at zero coupling.

Generalized Krefl-Schwarz representation for simply-laced algebras.

Derived representations for non-simply-laced algebras.

Abstract

We present a new expression for the partition function of refined Chern-Simons theory on $S^3$ with arbitrary gauge group, which is explicitly equal to $1$, when the coupling constant is zero. Using this form of partition function we show that the previously known Krefl-Schwarz representation of partition function of refined Chern-Simons theory on $S^3$ can be generalized to all simply-laced algebras. For all non-simply-laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with refined topological strings.